finite Fourier series

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1—10 of 13 matching pages

1: 27.10 Periodic Number-Theoretic Functions

… ►Every function periodic (mod

k

) can be expressed as a finite Fourier series of the form … ►is a periodic function of

n\pmod{k}

and has the finite Fourier-series expansion …

2: 29.20 Methods of Computation

… ►The corresponding eigenvectors yield the coefficients in the finite Fourier series for Lamé polynomials. …

4: 18.2 General Orthogonal Polynomials

… ►Let

(a,b)

be a finite or infinite open interval in

\mathbb{R}

. … ►Let

X

be a finite set of distinct points on

\mathbb{R}

, or a countable infinite set of distinct points on

\mathbb{R}

, and

w_{x}

x\in X

, be a set of positive constants. …when

X

is a finite set of

N+1

distinct points. …whereas in the latter case the system

\{p_{n}(x)\}

is finite:

n=0,1,\ldots,N

. … ►For such a system, functions

f\in L_{w}^{2}((a,b))

and sequences

\{\lambda_{n}\}

(

n=0,1,2,\ldots

) satisfying

\sum_{n=0}^{\infty}h_{n}|\lambda_{n}|^{2}<\infty

can be related to each other in a similar way as was done for Fourier series in (1.8.1) and (1.8.2): …

5: Bibliography T

… ►

A. Takemura (1984) Zonal Polynomials. Institute of Mathematical Statistics Lecture Notes—Monograph Series, 4, Institute of Mathematical Statistics, Hayward, CA.

ⓘ

External Links:: ISBN 0-940600-05-6, FullText, MathReview (A. W. Davis), ZentralBlatt
Referenced by:: §35.4(i), §35.9.
Encodings:: BibTeX

… ►

I. C. Tang (1969) Some definite integrals and Fourier series for Jacobian elliptic functions. Z. Angew. Math. Mech. 49, pp. 95–96.

ⓘ

External Links:: ISSN 0044-2267, Document, MathReview (B. C. Carlson), ZentralBlatt
Referenced by:: §22.11.
Encodings:: BibTeX

… ►

A. Terras (1999) Fourier Analysis on Finite Groups and Applications. London Mathematical Society Student Texts, Vol. 43, Cambridge University Press, Cambridge.

ⓘ

External Links:: ISBN 0-521-45718-1, MathReview (Stefan Kühnlein), ZentralBlatt
Referenced by:: §5.16.
Encodings:: BibTeX

… ►

E. C. Titchmarsh (1986a) Introduction to the Theory of Fourier Integrals. Third edition, Chelsea Publishing Co., New York.

ⓘ

Notes:: Original edition was published in 1937 by Oxford University Press. Expanded bibliography and some improvements were made for the second and third editions.
External Links:: ISBN 0-8284-0324-4, MathReview, ZentralBlatt
Referenced by:: §1.14(i), §1.14(ii), §1.14(iv), §1.14(v), §1.14(vii), §1.8(iv), §10.22(v), §10.32(iii), §5.13.
Encodings:: BibTeX

… ►

G. P. Tolstov (1962) Fourier Series. Prentice-Hall Inc., Englewood Cliffs, N.J..

ⓘ

Notes:: Translated from the Russian by Richard A. Silverman. Second English edition published by Dover Publications Inc., New York, 1976.
External Links:: MathReview (R. P. Boas, Jr.), ZentralBlatt
Referenced by:: §1.8(i), §1.8(ii).
Encodings:: BibTeX

…

6: 20.14 Methods of Computation

… ►The Fourier series of §20.2(i) usually converge rapidly because of the factors

q^{(n+\frac{1}{2})^{2}}

q^{n^{2}}

, and provide a convenient way of calculating values of

\theta_{j}\left(z\middle|\tau\right)

. … ►Hence the first term of the series (20.2.3) for

\theta_{3}\left(z\tau^{\prime}\middle|\tau^{\prime}\right)

suffices for most purposes. In theory, starting from any value of

\tau

, a finite number of applications of the transformations

\tau\to\tau+1

and

\tau\to-1/\tau

will result in a value of

\tau

with

\Im\tau\geq\sqrt{3}/2

; see §23.18. …

8: 1.17 Integral and Series Representations of the Dirac Delta

§1.17 Integral and Series Representations of the Dirac Delta

… ►

§1.17(ii) Integral Representations

►Formal interchange of the order of integration in the Fourier integral formula ((1.14.1) and (1.14.4)): … ►

§1.17(iii) Series Representations

►Formal interchange of the order of summation and integration in the Fourier summation formula ((1.8.3) and (1.8.4)): …

Originally it was stated that these Fourier series converge “…conditionally when $\nu$ is real and $0\leq\mu<\frac{1}{2}$ .” It has been corrected to read “If $0\leq\Re\mu<\frac{1}{2}$ then they converge, but, if $\theta\not=\frac{1}{2}\pi$ , they do not converge absolutely.”

Reported by Hans Volkmer on 2021-06-04

… ►

Subsection 25.2(ii) Other Infinite Series

It is now mentioned that (25.2.5), defines the Stieltjes constants $\gamma_{n}$ . Consequently, $\gamma_{n}$ in (25.2.4), (25.6.12) are now identified as the Stieltjes constants.

… ►

Section 1.14

There have been extensive changes in the notation used for the integral transforms defined in §1.14. These changes are applied throughout the DLMF. The following table summarizes the changes.

Transform	New	Abbreviated	Old
	Notation	Notation	Notation
Fourier	$\mathscr{F}\left(f\right)\left(x\right)$	$\mathscr{F}\mskip-3.0muf\mskip 3.0mu\left(x\right)$
Fourier Cosine	$\mathscr{F}_{\mkern-3.0muc}\left(f\right)\left(x\right)$	$\mathscr{F}_{\mkern-3.0muc}\mskip-1.0muf\mskip 3.0mu\left(x\right)$
Fourier Sine	$\mathscr{F}_{\mkern-2.0mus}\left(f\right)\left(x\right)$	$\mathscr{F}_{\mkern-2.0mus}\mskip-1.0muf\mskip 3.0mu\left(x\right)$
Laplace	$\mathscr{L}\left(f\right)\left(s\right)$	$\mathscr{L}\mskip-3.0muf\mskip 3.0mu\left(s\right)$	$\mathscr{L}(f(t);s)$
Mellin	$\mathscr{M}\left(f\right)\left(s\right)$	$\mathscr{M}\mskip-3.0muf\mskip 3.0mu\left(s\right)$	$\mathscr{M}(f;s)$
Hilbert	$\mathcal{H}\left(f\right)\left(s\right)$	$\mathcal{H}\mskip-3.0muf\mskip 3.0mu\left(s\right)$	$\mathcal{H}(f;s)$
Stieltjes	$\mathcal{S}\left(f\right)\left(s\right)$	$\mathcal{S}\mskip-3.0muf\mskip 3.0mu\left(s\right)$	$\mathcal{S}(f;s)$

Previously, for the Fourier, Fourier cosine and Fourier sine transforms, either temporary local notations were used or the Fourier integrals were written out explicitly.

►

Subsection 1.16(vii)

Several changes have been made to

(i)

make consistent use of the Fourier transform notations $\mathscr{F}\left(f\right)$ , $\mathscr{F}\left(\phi\right)$ and $\mathscr{F}\left(u\right)$ where $f$ is a function of one real variable, $\phi$ is a test function of $n$ variables associated with tempered distributions, and $u$ is a tempered distribution (see (1.14.1), (1.16.29) and (1.16.35));
(ii)

introduce the partial differential operator $\mathbf{D}$ in (1.16.30);
(iii)

clarify the definition (1.16.32) of the partial differential operator $P(\mathbf{D})$ ; and
(iv)

clarify the use of $P(\mathbf{D})$ and $P(\mathbf{x})$ in (1.16.33), (1.16.34), (1.16.36) and (1.16.37).

►

Subsection 1.16(viii)

An entire new Subsection 1.16(viii) Fourier Transforms of Special Distributions, was contributed by Roderick Wong.

…

10: 18.3 Definitions

… ►For finite power series of the Jacobi, ultraspherical, Laguerre, and Hermite polynomials, see §18.5(iii) (in powers of

x-1

for Jacobi polynomials, in powers of

x

for the other cases). Explicit power series for Chebyshev, Legendre, Laguerre, and Hermite polynomials for

n=0,1,\ldots,6

are given in §18.5(iv). … ►It is also related to a discrete Fourier-cosine transform, see Britanak et al. (2007). … ►For

-1-\beta>\alpha>-1

a finite system of Jacobi polynomials

P^{(\alpha,\beta)}_{n}\left(x\right)

is orthogonal on

(1,\infty)

with weight function

w(x)=(x-1)^{\alpha}(x+1)^{\beta}

. … ►However, in general they are not orthogonal with respect to a positive measure, but a finite system has such an orthogonality. …

finite Fourier series

1—10 of 13 matching pages

1: 27.10 Periodic Number-Theoretic Functions

2: 29.20 Methods of Computation

3: 1.8 Fourier Series

§1.8 Fourier Series

Uniqueness of Fourier Series

§1.8(ii) Convergence

4: 18.2 General Orthogonal Polynomials

5: Bibliography T

6: 20.14 Methods of Computation

7: 18.17 Integrals

§18.17(v) Fourier Transforms

Jacobi

Ultraspherical

Legendre

Hermite

8: 1.17 Integral and Series Representations of the Dirac Delta

§1.17 Integral and Series Representations of the Dirac Delta

§1.17(ii) Integral Representations

§1.17(iii) Series Representations

9: Errata

10: 18.3 Definitions